A Numerical Approach to Scalar Nonlocal Conservation Laws

TL;DR

This paper introduces a numerical algorithm for 1D scalar nonlocal conservation laws, demonstrates its convergence, and explores the properties and potential local limits of these nonlocal equations through computational experiments.

Contribution

The paper presents a new numerical method for nonlocal conservation laws and provides the first computational evidence of their properties and convergence behavior.

Findings

The algorithm converges reliably for nonlocal conservation laws.

Usual properties of local conservation laws may not hold in the nonlocal setting.

Numerical evidence suggests nonlocal equations may converge to local ones under certain conditions.

Abstract

We address the study of a class of 1D nonlocal conservation laws from a numerical point of view. First, we present an algorithm to numerically integrate them and prove its convergence. Then, we use this algorithm to investigate various analytical properties, obtaining evidence that usual properties of standard conservation laws fail in the nonlocal setting. Moreover, on the basis of our numerical integrations, we are lead to conjecture the convergence of the nonlocal equation to the local ones, although no analytical results are, to our knowledge, available in this context.